The arithmetic sequence $a_i$ is defined by the formula: $a_1 = -53$ $a_i = a_{i - 1} + 6$ Find the sum of the first $880$ terms in the sequence.
Answer: Getting started Let's write out the first few terms of the series: $-53 + (-47) + (-41) + (-35)...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $6$ greater than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-53})$ and the number of terms $(n = {880})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $880 -1= 879$ terms after the first term. The sequence increases by $6$ for each new term. So, the sequence increases by a total of $879 \cdot 6 = 5274$ from where it starts at $-53$. That means the last term must be $-53+5274 = {5221}$. In other words, $a_n = {5221}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{880}}&= \dfrac {\left({-53} + {5221} \right)}{2} \cdot {880} \\\\ S_{{880}} &= 2584 \left(880\right) \\\\ S_{{880}} &= 2{,}273{,}920\end{aligned}$ The answer $ 2{,}273{,}920 $